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Points of quantum $\mathrm{SL}_n$ coming from quantum snakes

Daniel C. Douglas

TL;DR

This work establishes that the quantized FG monodromy matrices associated to triangles and edges satisfy the relations of the quantum group SL$_n^q$, by developing a quantum analogue of Fock–Goncharov’s snakes. The authors construct left and right quantum matrices, prove they are SL$_n^q$-points in a quantum torus, and prove this by a snake-sweep factorization into commuting subalgebras embedded in a tensor product of snake-move algebras. A key technical novelty is the embedding of a distinguished subalgebra T$_L$ into a tensor product of snake-move algebras, which allows the global SL$_n^q$-property to be deduced from local, elementary SL$_n^q$-factors. This result provides a local, algebraic cornerstone toward a conceptual quantum trace map for knots in thickened surfaces and connects higher Teichmüller theory with quantum group theory, with broader implications for quantum topology and noncommutative geometry.

Abstract

We show that the quantized Fock-Goncharov monodromy matrices satisfy the relations of the quantum special linear group $\mathrm{SL}_n^q$. The proof employs a quantum version of the technology invented by Fock-Goncharov called snakes. This relationship between higher Teichmüller theory and quantum group theory is integral to the construction of a $\mathrm{SL}_n$-quantum trace map for knots in thickened surfaces, partially developed in a companion paper (arXiv:2101.06817).

Points of quantum $\mathrm{SL}_n$ coming from quantum snakes

TL;DR

This work establishes that the quantized FG monodromy matrices associated to triangles and edges satisfy the relations of the quantum group SL, by developing a quantum analogue of Fock–Goncharov’s snakes. The authors construct left and right quantum matrices, prove they are SL-points in a quantum torus, and prove this by a snake-sweep factorization into commuting subalgebras embedded in a tensor product of snake-move algebras. A key technical novelty is the embedding of a distinguished subalgebra T into a tensor product of snake-move algebras, which allows the global SL-property to be deduced from local, elementary SL-factors. This result provides a local, algebraic cornerstone toward a conceptual quantum trace map for knots in thickened surfaces and connects higher Teichmüller theory with quantum group theory, with broader implications for quantum topology and noncommutative geometry.

Abstract

We show that the quantized Fock-Goncharov monodromy matrices satisfy the relations of the quantum special linear group . The proof employs a quantum version of the technology invented by Fock-Goncharov called snakes. This relationship between higher Teichmüller theory and quantum group theory is integral to the construction of a -quantum trace map for knots in thickened surfaces, partially developed in a companion paper (arXiv:2101.06817).

Paper Structure

This paper contains 43 sections, 12 theorems, 93 equations, 21 figures.

Table of Contents

  1. Fock-Goncharov snakes
  2. Generic configurations of flags and Fock-Goncharov invariants
  3. Generic triples and quadruples of flags
  4. Discrete triangle
  5. Fock-Goncharov triangle and edge invariants
  6. Action of $\mathrm{PGL}(V)$ on generic flag triples
  7. Snakes and projective bases
  8. Snakes
  9. Line decomposition of $V^*$ associated to a generic triple of flags and a snake
  10. Projective basis of $V^*$ associated to a generic triple of flags and a snake
  11. Snake moves
  12. Elementary matrices
  13. Adjacent snake pairs
  14. Diamond and tail moves
  15. Right snakes and right snake moves
  16. ...and 28 more sections

Key Result

Theorem 1

When $C$ is an arc on the corner of a triangle $\lambda_k$, the Fock-Goncharov quantum matrix $\mathbf{M}^q_C \in \mathrm{M}_n(\mathscr{T}_n^q(\lambda_k))$ is a $\mathscr{T}_n^q(\lambda_k)$-point of the quantum special linear group $\mathrm{SL}_n^q$. In other words, each such matrix defines an algeb by the property that the $n^2$-many generators of the algebra $\mathrm{SL}_n^q$ are sent to the cor

Figures (21)

  • Figure 1: Discrete triangle, and triangle invariants for a generic flag triple
  • Figure 2: Edge invariants for a generic flag quadruple
  • Figure 3: Snake
  • Figure 4: Three coplanar lines involved in the definition of a projective basis. For the meaning of the $+$ and $-$ signs, see Definition \ref{['def:defining-the-projective-basis-by-induction']}.
  • Figure 5: Diamond move
  • ...and 16 more figures

Theorems & Definitions (45)

  • Theorem
  • Definition 1.1
  • Theorem 1.2: Fock-Goncharov
  • proof
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Proposition 1.8: Fock-Goncharov
  • ...and 35 more